Extendability of the $B_2$-arrangement

Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always exists. In this work, we demonstrate that if $\mathscr{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathscr{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank.